Assume towards a contradiction that \(|\partial {\Omega }|>0\) . By the Lebesgue density point theorem and le:setFbound, there exists a measurable subset \(U\)  of \(\partial {\Omega }\)  with \(|U|=|\partial {\Omega }|\)  such that every \(x\in U\)  is a Lebesgue point of \(\partial {\Omega }\)  and \(\overline{D\Phi }(x)<\infty \) . Fix \(x\in U\) . Since \(x\)  is a Lebesgue point, there exists a sufficiently small \(r_x>0\) , such that for every \(0<r<r_x\) , we have
\(\left|B(x, r)\cap \overline{{\Omega }}\right|\ge \frac{1}{2^{n-1}}\left|B(x, r)\right|.\) 
Let \(r\in (0, r_x)\)  be fixed. Since \(|\partial B(x, s)|=0\)  for every \(s\in (0, r)\) , we have
\(\left|B\left(x, \frac{r}{4}\right)\cap \overline{{\Omega }}\right|\ge \frac{1}{2^{n-1}}\left|B\left(x, \frac{r}{4}\right)\right|\ge \frac{1}{2^{3n-1}}\left|B(x, r)\right|\) 
and
\(\left|\left(B(x,r)\setminus B\left(x, \frac{r}{2}\right)\right)\cap \overline{{\Omega }}\right|\ge |B(x, r)\cap \overline{{\Omega }}|-\left|B\left(x, \frac{r}{2}\right)\right|\ge \frac{1}{2^n}|B(x, r)|.\) 
Define a test function \(u\in W^{1, p}({\Omega })\cap C({\Omega })\)  by setting
\(u(y):={\left\lbrace \begin{array}{ll}1,\ \ &{\rm if}\ y\in B\left(x, \frac{r}{4}\right)\cap {\Omega },\\\frac{-4}{r}|y-x|+2,\ \ &{\rm if}\ y\in \left(B\left(x, \frac{r}{2}\right)\setminus B\left(x, \frac{r}{4}\right)\right)\cap {\Omega },\\0,\ \ &{\rm if}\ y\in {\Omega }\setminus B\left(x, \frac{r}{2}\right).\end{array}\right.}\) 
We have
\(\left(\int _{B(x, r)\cap {\Omega }}|u(y)|^p+|\nabla u(y)|^p\,dx\right)^{\frac{1}{p}}\le \frac{C}{r}|B(x, r)\cap {\Omega }|^{\frac{1}{p}}.\) 
Since \(u\equiv 0\)  on \({\Omega }\setminus B(x, r/2)\) , we have \(u\in W^p_0(B(x, r), {\Omega })\) . Then, by the definition (REF ) of the set function \(\Phi \)  and by cor:extension, there exists a function \(v\in BV(B(x, r))\)  with \(v\big |_{B(x, r)\cap {\Omega }}\equiv u\)  and
\(\Vert Dv\Vert (B(x, r))\le 2\Phi ^{\frac{1}{k}}(B(x, r))\Vert u\Vert _{W^{1, p}(B(x, r)\cap {\Omega })}.\) 
By the Poincaré inequality of \(BV\)  functions stated in prop:BVpoin, we have
\(\int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\le C r\Vert Dv\Vert (B(x, r)).\) 
Since \({\Omega }\)  is 1-fat on almost every \(z\in \partial {\Omega }\) , by le:fat, \(v(z)=1\)  for almost every \(z\in B\left(x,\frac{r}{4}\right)\cap \partial {\Omega }\)  and \(v(z)=0\)  for almost every \(z\in \left(B(x,r)\setminus B\left(x, \frac{r}{2}\right)\right)\cap \partial {\Omega }\) . Hence, on one hand, if \(v_{B(x, r)}\le \frac{1}{2}\) , we have
\(\int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\ge \frac{1}{2}\left|B\left(x, \frac{r}{4}\right)\cap \overline{{\Omega }}\right|\ge c|B(x, r)|.\) 
On the other hand, if \(v_{B(x, r)}>\frac{1}{2}\) , we have
\(\int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\ge \frac{1}{2}\left|\left(B(x, r)\setminus B\left(x, \frac{r}{2}\right)\right)\cap \overline{{\Omega }}\right|>c|B(x, r)|.\) 
All in all, we always have
\(\int _{B(x, r)}|v(y)-v_{B(x, r)}|\,dy\ge c|B(x, r)|\) 
for a sufficiently small constant \(c>0\) . Thus, by combining inequalities (REF )-(REF ), we obtain
\(\Phi (B(x, r))^{p-1}|B(x, r)\cap {\Omega }|\ge c|B(x, r)|^p\) 
for a sufficiently small constant \(c>0\) . This gives
\(|B(x, r)\cap \partial {\Omega }|\le |B(x, r)|-|B(x, r)\cap {\Omega }|\le |B(x, r)|-C\frac{|B(x, r)|^p}{\Phi (B(x, r))^{p-1}}.\) 
Since \(\overline{D\Phi }(x)<\infty \) , we have
\(\limsup _{r\rightarrow 0^+}\frac{|B(x, r)\cap \partial {\Omega }|}{|B(x, r)|}&\le &\limsup _{r\rightarrow 0^+}\left(1-\frac{|B(x, r)\cap {\Omega }|}{|B(x, r)|}\right)\nonumber \\&\le &\limsup _{r\rightarrow 0^+}\left(1-\frac{|B(x, r)|^{p-1}}{\Phi (B(x, r))^{p-1}}\right)\le 1-c\overline{D\Phi }(x)^{1-p}<1.\nonumber \) 
This contradicts the assumption that \(x\)  is a Lebesgue point of \(\partial {\Omega }\) . Hence, we conclude that \(|\partial {\Omega }|=0\) .
Let us then consider the case \({\Omega }\subset {\mathbb {R}}^2\) . By [1]}, for every \(x\in \partial {\Omega }\)  and every \(0<r<\min \left\lbrace 1, \frac{1}{4}{\mathop \mathrm {\,diam\,}}({\Omega })\right\rbrace \) , we have
\(Cap_1({\Omega }\cap B(x, r); B(x, 2r))\ge cr\) 
for a constant \(0<c<1\) . This implies that \({\Omega }\)  is 1-fat at every \(x\in \partial {\Omega }\) . Hence, by combining this with the first part of the theorem, we have that the boundary of any planar \((W^{1, p}, BV)\) -extension domain is of volume zero.
